inverse of f(x)=2+1/(x-4)
1) Let's sum the fractions:
ⓘ To add fractions, put them over the same denominator, then add the numerators: =
ⓘ Expand by multiplying each term inside the parentheses: =
ⓘ The numerator and the denominator are computed before the division: =
2) For a function , two inputs u and v with the same output satisfy after multiplying by the denominators. Here, = ≠ 0, so u = v. The function therefore has an inverse on its range
Express input in terms of output :
⇥2.1) Let's solve in :
⇥2.2) Let's add fractions:
⇥ⓘ To add fractions, put them over the same denominator, then add the numerators: =
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⇥A quotient is zero if and only if its numerator is zero and its denominator is not: we solve the numerator equal to 0
⇥⇥2.2.1) Let's solve in :
⇥⇥ⓘ Expand by multiplying each term inside the parentheses: =
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⇥⇥ⓘ A minus sign before a parenthesis changes the sign of each of its terms: =
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⇥⇥ⓘ Two minus signs cancel out: =
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⇥⇥ⓘ Both sides are divided by , which must be nonzero:
⇥⇥Solution found:
⇥⇥2.2.2) To prevent the denominator from being zero, let's solve:
⇥⇥Domain condition:
Renaming the output as , the inverse is , with domain , the condition found when isolating the input
▶Inverse function:
